I have a little doubt. These quantities
$\inf_{n} f_n $ , $\sup_n f_n$
are function only if the there exist the infimum of {$f_n(x)$} $\forall x $ and is different from $\infty$.
And the same is for the superior and inferior limit.
Am I right?
For instance, for the sequence of constant function $f_n(x) = n$ we can't define $\sup_n f_n $ right?